- Interval exchange transformation
In
mathematics , an interval exchange transformation is a kind ofdynamical system that generalises the idea of acircle rotation . The phase space consists of theunit interval , and the transformation acts by cutting the interval into several subintervals, and then permuting these subintervals.Formal definition
Let n > 0 and let pi be a
permutation on 1, dots, n. Consider a vector:lambda = (lambda_1, dots, lambda_n)
of positive real numbers (the widths of the subintervals), satisfying
:sum_{i=1}^n lambda_i = 1.
Define a map
:T_{pi,lambda}: [0,1] ightarrow [0,1] ,
called the interval exchange transformation associated to the pair pi,lambda) as follows. For
:1 leq i leq n
let
:a_i = sum_{1 leq j < i} lambda_j and let
:a'_i = sum_{1 leq j < pi(i)} lambda_{pi^{-1}(j)}.
Then for x in [0,1] , define
:T_{pi,lambda}(x) = x - a_i + a'_i
if x lies in the subinterval a_i,a_i+lambda_i). Thus T_{pi,lambda} acts on each subinterval of the form a_i,a_i+lambda_i) by an orientation-preserving
isometry , and it rearranges these subintervals so that the subinterval at position i is moved to position pi(i).Properties
Any interval exchange transformation T_{pi,lambda} is a
bijection of 0,1] to itself which preservesLebesgue measure . It is not usually continuous at each point a_i (but this depends on the permutation pi).The
inverse of the interval exchange transformation T_{pi,lambda} is again an interval exchange transformation. In fact, it is the transformation T_{pi^{-1}, lambda'} where lambda'_i = lambda_{pi^{-1}(i)} for all 1 leq i leq n.If n=2 and pi = (12) (in
cycle notation ), and if we join up the ends of the interval to make a circle, then T_{pi,lambda} is just a circle rotation. TheWeyl equidistribution theorem then asserts that if the length lambda_1 isirrational , then T_{pi,lambda} isuniquely ergodic . Roughly speaking, this means that the orbits of points of 0,1] are uniformly evenly distributed. On the other hand, if lambda_1 is rational then each point of the interval is periodic, and the period is the denominator of lambda_1 (written in lowest terms).If n>2, and provided pi satisfies certain non-degeneracy conditions, a deep theorem due independently to W.Veech and to H.Masur asserts that for
almost all choices of lambda in the unit simplex t_1, dots, t_n) : sum t_i = 1} the interval exchange transformation T_{pi,lambda} is againuniquely ergodic . However, for n geq 4 there also exist choices of pi,lambda) so that T_{pi,lambda} isergodic but notuniquely ergodic . Even in these cases, the number of ergodicinvariant measures of T_{pi,lambda} is finite, and is at most n.References
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